The easy way
There is a very easy way to do so:
You can use
Geoprocessing Tools ->
Input Layer is the underlying, greenish one.
Overlay Layer is the clipping, orange one.
This will create a new layer (and if you leave the
Clipped name empty, it will default to a temporary, disposable layer, so you don't pollute your hard drive).
The number of individual features of the clipped layer is the number you are looking for.
The flexible way
You can avoid doing the manual count (useful if you have a large number of features) by using:
Toolbox (the "gear" icon) ->
Join attributes by location (summary)
and inputting something like in the following screenshot:
Please note that, depending on the options you select, you can yield many more useful insights about the newly clipped polygons ;)